Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem

Chiarolla MB, Ferrari G (2014)
SIAM Journal on Control and Optimization 52(2): 1048-1070.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Chiarolla, Maria B.; Ferrari, GiorgioUniBi
Abstract / Bemerkung
We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Ito diffusion controlled by a non-decreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality, and we characterize the optimal solution in terms of the base capacity process l*(t), i.e., the unique solution of a representation problem in the spirit of Bank and El Karoui [P. Bank and N. El Karoui, Ann. Probab., 32 (2004), pp. 10301067]. We show that the base capacity is deterministic and it is identified with the free boundary y(t) of the associated optimal stopping problem when the coefficients of the controlled diffusion are deterministic functions of time. This is a novelty in the literature on finite horizon singular stochastic control problems. As a subproduct this result allows us to obtain an integral equation for the free boundary, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process.
Stichworte
capacity; irreversible investment; singular stochastic control; base; free boundary; optimal stopping; Bank and El Karoui's Representation Theorem
Erscheinungsjahr
2014
Zeitschriftentitel
SIAM Journal on Control and Optimization
Band
52
Ausgabe
2
Seite(n)
1048-1070
ISSN
0363-0129
eISSN
1095-7138
Page URI
https://pub.uni-bielefeld.de/record/2681888

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Chiarolla MB, Ferrari G. Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem. SIAM Journal on Control and Optimization. 2014;52(2):1048-1070.
Chiarolla, M. B., & Ferrari, G. (2014). Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem. SIAM Journal on Control and Optimization, 52(2), 1048-1070. doi:10.1137/11085195X
Chiarolla, M. B., and Ferrari, G. (2014). Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem. SIAM Journal on Control and Optimization 52, 1048-1070.
Chiarolla, M.B., & Ferrari, G., 2014. Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem. SIAM Journal on Control and Optimization, 52(2), p 1048-1070.
M.B. Chiarolla and G. Ferrari, “Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem”, SIAM Journal on Control and Optimization, vol. 52, 2014, pp. 1048-1070.
Chiarolla, M.B., Ferrari, G.: Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem. SIAM Journal on Control and Optimization. 52, 1048-1070 (2014).
Chiarolla, Maria B., and Ferrari, Giorgio. “Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank--El Karoui Representation Theorem”. SIAM Journal on Control and Optimization 52.2 (2014): 1048-1070.